The techniques developed in this manuscript, are an improvement with respect to the existing ones, in the sense that developed a unified framework for solutions of Boltzmann flows for binary elastic interactions, for hard potential and integrable differential crossections conforming the transition probabilities in such interacting particle flows.
The specifics developed here can be applied to any solution \(f(v,t) \in \Omega \)  of the Boltzmann equation constructed by Theorem REF , were inspired in the works on [1]}, [2]}, [3]}, [4]}, [5]} for elastic and inelastic theories, by framing the summability of moments into shifted power series (or Mittag-Leffler form), as well as on the work of [6]} introducing the technique of study the summability of partial sums for hard potentials with integrable angular part of the transition probability, and by non-integrable cross sections in [7]}, [8]} for Maxwell type of interactions with or without the angular integrability requirement.
